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Comment by amscanne | Hacker News Reader

Love it. The answer is 1/2. The first event occurs with a population of 2,2 and the second event has no causal effect. The trap is that the observation doesn’t change the sample population of the first event, exactly like the Monty Hall case: opening doors itself does not change the probability to 1/2 as people think it does (it remains 1/3) — it’s the switch that changes the odds.

Similar question with a similar trap: there are new neighbors moving in next door, and you know they have two kids. You see a boy in their yard, so you know they have at least one boy. What’s the probability they have two boys?

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Take a look at the other replies. I like paxys' for the simplicity.

The second ball being selected doesn't change the first event, but it does change our understanding of it.

An extreme version: There's a bowl with 3 red and 1 blue balls. We remove two balls again, and the second one is blue. What are the odds that the first one is blue?

Your two kids problem is actually pretty complex, in the form you phrased it. Wiki has a decent explanation (I contend that your question is equivalent to the second question in the wiki article): https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

I’m aware that it’s complicated. I intentionally phrased it in a way that makes it clear that the family moving in is not somehow selected from the set of families with at least one boy, but rather the observation is unrelated.

This is the distinction. I believe that the 1/3 analysis may also be incorrect for the way you phrased your question. If you had said: “we select a second ball, and only observe the first ball if the second is blue — what are the chances of it being blue?”, then the second observation controls the population. Otherwise it’s semantics over exactly what probability we are trying to define?

I think the issue with these questions is that we are asking about “probabilities” which only make sense with repeated iterations. So you always have ambiguity in the construction of the question and interpretation when asking about things that are a “one time” event like both these examples.

I don't think this is correct. It matters whether the person removing the second ball can see the colors and choose accordingly. If you can, and you willingly remove a single blue ball, you have the Monty hall problem where you stick to your choice. It does not influence the chances, it is still 1/2. But the way I read the problem, you choose a ball at random, look at its color, it happens to be blue. Now this gives you information on the composition of the remaining balls. The chance is 1/3. To drive home the point that removing a ball "can influence" the odds of the first ball: what if you took out an extra ball that turns out to be blue? You now have two blue balls, leaving the chance the first ball was blue zero.

The sneaky part of this question is that it gives information about the outcome and then asks you again about the probability. Forget the extra balls, and just drive the point home directly: suppose I pick a ball at random, and show it to you. It’s blue. I ask: “what are the odds that the ball is blue?” (exact phrasing as original question).

There are obviously correct interpretations for 50% and 100%. It depends whether I’m asking: - What’s the probability of this outcome? - What’s the probability that the ball I’m holding in my hand in blue?

The second is effectively a “resampling” with a population of one. You are simply assuming the second interpretation and arguing for it, but I don’t dispute the logic. The original question is unclear whether it’s asking for the probability that you picked a blue ball initially (50%) or the likelihood that the ball is blue, given some information of the outcome. But we don’t normally speak of probabilities this way. The odds that you picked a blue ball initially were 50%, even if you picked a red one.

By giving only partial information, the question creates more ambiguity since the answer isn’t definite. (When there is ambiguity in a question, I believe most people will discard trivial interpretations over substantive ones, which is what pushes toward the “resample” here.)

Anyway, I’ll leave it there since I think it’s clear there are correct interpretations for both, depending on what the question is actually asking.

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